Proceedings of the International Conference on Advances in Control and Optimization of Dynamical Systems (ACODS2007)
Abstract— This paper addresses the problem of multiagent search in an unknown environment. The agents are autonomous in nature and are equipped with necessary sensors to carry out the search operation. The uncertainty, or lack
of information about the search area is known a priori as a probability density function. The agents are deployed in an optimal way so as to maximize the one step uncertainty reduction. The agents continue to deploy themselves and reduce uncertainty till the uncertainty density is reduced over the search space below a minimum acceptable level. It has been shown, using LaSalle’s invariance principle, that a distributed control law which moves each of the agents towards the centroid of its Voronoi partition, modified by the sensor range leads to single step optimal deployment. This principle is now used to devise search trajectories for the agents. The simulations were carried out in 2D space with saturation on speeds of the agents. The results show that the control strategy per step indeed moves the agents to the respective centroid and the algorithm reduces the uncertainty distribution to the required level within a few steps.
I. INTRODUCTION
Search in an unknown environment using mobile sensor networks or UAVs equipped with sensors is addressed in this paper. Agents equipped with required sensors can be deployed in large scale for a wide variety of applications such as search and rescue, environmental monitoring, military and defence application etc. Multi-agent search using path planning, game theoretic, and behavior-based approach have been reported in the literature. A comprehensive literature on multi-agent search can be found in [1] and [2]. In
[3] and [4] authors use the concept of centroidal Voronoi configuration for optimal deployment of multiple agents in a convex hull with known probability density function. These results will be used as bases for our work.
In this paper, optimal deployment of, and search operation by, multiple agents in an unknown environment is considered. The search space is a convex polytope
Q and the initial uncertainty density distribution is assumed to be known a priori.
N agents equipped with sensors and communication equipments, deploy themselves in Q , and gather information, thus reducing the uncertainty. The deployment leading to one step optimal information gathering is achieved by centroidal Voronoi configurations which is modified by the sensor range. The agents continue to deploy themselves optimally in Q with updated uncertainty density
Guruprasad K. R. is Lecturer at Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, India and is presently pursuing doctoral studies at Guidance, Control, and Decision
Systems Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India. email: krguru@aero.iisc.ernet.in
Debasish Ghose is Professor at Guidance, Control, and Decision Systems Laboratory, Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India. email: dghose@aero.iisc.ernet.in
© ACODS2007 distribution and gather information till the uncertainty is reduced throughout Q below a minimum acceptable level.
The organization of this paper is as follows. In Section II the problem formulation is discussed, and critical points of the objective functions are derived. A proportional control law is proposed in Section III and is shown to be globally asymptotically stable using LaSalle’s invariance principle.
Section IV presents the simulation results and Section V concludes the paper.
II. M ULTI AGENT SEARCH
N agents perform search operation in an unknown environment. The lack of information is modeled as an uncertainty density distribution over the search space Q .
The problem addressed in this paper is of deploying the
N agents in Q to collect information, thereby reducing the uncertainty density distribution over Q . The problem formulation is stated formally as
•
•
•
•
•
Q ⊂ R n is a convex polytope and is the search space
φ : Q 7→ [0 , 1] defines the density function representing uncertainty (lack of information)
N agents equipped with sensors and communication equipments, deploy themselves in Q , and gather information, thus reducing the uncertainty.
Sensor’s effectiveness reduces with Euclidean distance.
Ideally, we are looking for an optimal way of utilizing the agents to acquire complete information about Q , and thus have φ ( q ) = 0 , ∀ q ∈ Q
At each iteration, after deploying themselves optimally, the sensors gather information about
Q
, reducing the density function as,
φ n +1
( q ) = φ n
( q ) min i
{ β ( k x i
− q k ) }
(1) where φ n
β : R
+
( q ) is the density function at the
7→ [0 , 1] is the factor of reduction in uncertainty by the sensors and, x i is the position of the i n -th iteration,
-th sensor. At a given q ∈ Q
, only the agent with the smallest
β ( k x i
− q k )
, that is, agent which can reduce the uncertainty by the largest amount is active. It is clear that
β ∈ [0 , 1]
.
A. Objective function (One-Step)
We are looking for deployment of agents in Q , maximizing per iteration reduction in the uncertainty φ . Consider the following objective function to be maximized.
380

H n
=
R
Q
∆ φ n
( q ) dq
=
=
R
Q max i
{ ( φ n
( q ) − β ( k x i
− q k ) φ n
( q )) } dq
R
Q
( φ n
( q ) − min i
{ β ( k x i
− q k ) } φ n
( q )) dq
=
P i
R
V i
φ n
( q )(1 − β ( k x i
− q k )) dq
(2) where
V i is the Voronoi partition (Figure 1) corresponding to i
-th agent, and x i
∈ Q is the position of the i
-th agent.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 ke
−
α
r
2
β
= 1 − ke
−
α
r
2 k = 0.8
α
= 0.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
0.9
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
1
1
Fig. 1.
Voronoi partitions for a set of points
The gradient is given by [3]
∆ H n
δx i
=
Z
V i
φ n
( q )
∂
∂x i
(1 − β ( r )) dq where r = k x i
− q k
(3)
B. Selection of β
β : R 7→ [0 , 1] is a non-decreasing function capturing effectiveness of the sensor. A continuously differentiable function leads to a possible closed form solution to the optimization problem (2). Consider
2 4 r
6 8
Fig. 2.
The sensor sensitivity and updation function
10
C. Optimal solution
The objective function (2) will now take the form
H n
=
X
Z i
V i
φ n
( q ) ke
− αr
2 dq
The gradient w.r.t.
x i is,
(4)
∂ H
∂x i n =
P i
R
V i
φ n
( q ) ke − α ( k x i
− q k )
2
( − 2 α )( x i
− q ) dq
= − 2 α
˜
V i
( x i
−
V i
)
(5)
Where
V i centroid of V i and
V i are respectively the mass and the with respect to φ 0 n
( q ) = φ n
( q ) ke − αr
2
, the density function as perceived by the sensor.
Thus the necessary condition for optimality is, x i V i
(6)
Note that
V i in a fixed point of depends on x i
V i
( x i
)
, that is, we are interested
III. T HE CONTROL LAW
Let us consider the system dynamics as
β ( r ) = 1 − ke − αr
2
, k ∈ (0 , 1)
Here, ke − αr
2 represents the sensitivity of the sensor which is maximum at r = 0 and tends to zero as r → ∞ and β is minimum at r = 0 (effecting maximum reduction in φ ) and tends to unity as r → ∞ (Figure 2) (change in φ reduces to zero as r increases) x
Consider the control law i
= u i
(7) u i
= − k prop
( x i
−
V i
) (8)
Control law (8) moves the agent towards k prop
381
.
V i for positive

A. Stability (LaSalle’s invariance principle)
Consider the
V ( X ) = −H n
IV. R
, where
X represents the configuration of
N agents.
= ( x
ESULTS AND DISCUSSIONS
1
, x
2
, ..., x
N
)
˙
( X ) = − d H n dt
= −
P i
= 2 α
P i
δ H
δx n i x i
V
I
( x i
−
= − 2 αk prop
P i
V i
)( − k prop
V i
( x i
−
V i
) 2
( x i
− C 0
V i
))
(9)
We can observe that
1) V : Q 7→ R is continuously differentiable in Q
2) M = Q is compact invariant set
3)
4)
V
E is negative definite in
= V − 1 (0) = {
V i
}
M
5) N = E , the largest invariant subset of E by the control law (8)
Thus by LaSalle’s invariance principle, the trajectories of the agents governed by control law (8), starting from any initial configuration
X (0) ∈ Q N
, will asymptotically converge to set
N
, the critical points of
H n
, i.e., the centroidal Voronoi partitions with respect to the density function as perceived by the sensors.
The algorithm implemented was Deploy and then search.
The agents are deployed optimally, and at the end of deployment step, the search task is performed. Each deployment
and search step will be referred to as one iteration. The iterations are continued till the average density is reduced below the required level.
A simulation experiment is carried out to test search algorithm using Matlab. Multi-parametric tool box [5] was used for functions related to Voronoi partitioning. Following are the parameters used in the simulation
•
•
•
•
•
•
•
N = 5
Q is a square area in R 2 with axes range of 0-10
Initial uncertainty density was a constant distribution of 0.75 over Q
A saturation on the movement of agents was fixed at 1 k prop
= 0 .
5
Sensor parameters, k = 0 .
8 and
α = 0 .
1
The iterations were terminated when the average density over
Q reached below 0.05
The simulations lasted for 5 iterations. Figure 3 shows the trajectory of agents in one iteration moving towards the centroid from the initial location, and final Voronoi diagram.
The trajectories of all 5 agents and the Voronoi diagram corresponding to final agent locations is shown in Figure 4.
The initial and final uncertainty distributions are shown in
Figure 5.
7
6
5
4
3
2
1
0
0
10
9
8
1 2 3 4 5 x
1
6
+ Starting point o centroid
7 8 9 10
Fig. 3.
Trajectories of the agents and the final Voronoi diagram during initial deployment step.
It can be seen that in spite of discrete implementation and use of saturation, the control strategy achieves the goal and successfully reduces the uncertainty density distribution.
It is observed during simulation experiments conducted for different conditions, that the strategy proposed in this paper performs fairly well. Though not encountered during the simulation experiments, it can not be ruled out that at some point of time, between successive iterations, the centroids do not change due to some kind of symmetry in the density distribution and agent distribution. That is, the search step modifies the uncertainty distribution in such a way that, even with modified density distribution, the centroids of none of the Voronoi partitions change. As a simple example, consider a single agent with initial constant density distribution. If the agent is at the center of the area, then it will stay in the center throughout. In such a case, the symmetry can be broken by giving small random motions to the agents when they get stuck in a place before reaching the termination condition.
V. C ONCLUSION AND FUTURE WORK
The problem of multi-agent search in an unknown environment with a known probability distribution function is formulated. In each step, the agents are deployed in an optimal way so as to maximize the one-step reduction in uncertainty and then the agents gather information in their respective Voronoi partitions and hence reduce the uncertainty. The iterations can be continued till the uncertainty in the entire region is reduced to a required level.
It has been shown that the one-step optimal deployment is the centroidal Voronoi configuration modified by the sensor range. A control law achieving the optimal deployment is proposed. It has also been shown using LaSalle’s invariance
382

7
6
5
4
3
2
1
0
0
10
9
8
1 2
Agent trajectotris & Final Voronoi diagram
Initial Uncertainty density
3 4 5 x
1
6
The density distibutions
7 8 9 10
Fig. 4. Trajectories of all 5 agents during 5 iterations and the final Voronoi diagram. ’+’ indicate the initial position of agents needs to be addressed. A continuous search algorithm, as opposed to deploy and then search implemented here, can be explored and the their performances can be compared.
The related control strategies will be studied and an optimal strategy taking into consideration more realistic constraints such as fuel limitations, time restrictions and computation requirements will be addressed in future work. It is also proposed to look into other possible uncertainty density updation strategies and compare their performances over the present strategy.
R EFERENCES
[1] P.B. Sujit and D. Ghose, ”Multiple UAV search using agent based negotiation scheme”, Proc. of American Control Conference, June
8-10, 2005. Portland, OR, USA, pp. 2995-3000
[2] S. Poduri and G. S. Sukhatme ”Constrained coverage for mobile sensor networks”, Porc. of IEEE International Conference on Robotics
and Automation, New Orleans, LA, April 2004, pp. 165-171
[3] J. Cortes, S. Martinez, T. Karata, and F. Bullo, ”Coverage control for mobile sensing networks”, IEEE Transactions on Robotics and
Automation, vol 20, no. 2, pp. 243-255, 2004
[4] J. Cortes, S. Martinez, and F. Bullo, ”Spatially-distributed coverage optimization and control with limited-range interactions”, ESAIM:
Control, Optimization and Calculus of Variations 11 (4) pp. 691-
719, 2005
[5] Multi-Parametric Toolbox for Matlab by M. Kvasnica, P. Grieder, and M. Baoti (http://control.ee.ethz.ch/ mpt/)
0.8
0.6
0.4
0.2
0
10
5
Final Uncertainty density
Y
0 0
2
4
X
6
8
10
Fig. 5.
Initial and final uncertainty distribution principle that the proposed control law achieves the one-step optimal deployment.
Simulation experiments are conducted for different conditions and results of one such experiment is discussed. The simulation results prove that the proposed search strategy performs quite well.
It can be noted that in present work, it is assumed that the network is connected and the sensors do no have any limitation on their range. It is proposed to relax these assumptions in future work. The result of one-step optimality needs to be extended and optimality of the overall search operation
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